The problem of approximating \((,)\)  with \((_N,_N)\) , where we take \((_N,_N)\)  to be the naive plugin estimators is studied in [1]}, [2]} for the balanced setting, and in [3]} for unbalanced OT.
When \(=^d\) , those results break the curse of dimensionality by transferring the exponential dependency on the dimension \(d\)  from the number of samples \(N\)  to the constants.
Assuming \((,)\)  are sub-Gaussian distributions and \(x,y)={x-y}^2_2\)  yields a rate \([|(_N,_N)-(,)|]=O((\epsilon ^{-\lceil 5d/4\rceil + 3} + 1)N^{-1/2})\)  in [2]}.
The works [5]}, [3]} prove a similar rate under the assumption that \((,)\)  are compactly supported measures, for general smooth cost \(.The result of~\cite {sejourne2019sinkhorn} is an extension of~\cite {genevay2018learning} to unbalanced OT.We summarize their results in the following theorem.\) 
Assume \(\phi ^*\)  and \( are \)\(, and that \) (,)\( are supported on a compact set. Then one has{\begin{@align*}{1}{-1}[|(_N,_N)-(,)|]=O((\epsilon ^{-\lfloor d/2\rfloor } + 1)N^{-1/2}).\end{@align*}}\)
